On Thu, Aug 30, 2012 at 7:03 AM, Peter Duveen <pduveen@yahoo.com> wrote:> So my point, I guess, is that, for the purposes of teaching, there is much to be gained from a knowledge of the historical sequence of mathematical development.

I completely agree with this.

The Egyptian style of expressing fractions appears to have hadeverything to do with apportionment of grains through containers offixed volume. Along the Nile, with fields but a thin margin on eitherside, such apportionment was life itself (its circulatory system).

These days, efficiently packing a truck and routing it, such thatpackages in need of off-loading aren't buried in the back. UPS truckshave shelves for random access right? That way routing and packingstay two different problems. But some loads are too bulky forshelving.

Continued fractions are another area that, along with much of numbertheory, has fallen by the wayside in a youth's education. Yet theseare the ideal little challenges one looks for when learning acomputer language. More fun than inverting a matrix, or in the samecategory at least, with opportunities to use recursion. You get toscrunch up your brain and do something precise. Then, unlike withpaper and pencil, you actually get a self-running algorithm more oftenthan not. Something you wrote. Something you might reuse.

Of course with just your plain old everyday interactive prompt you canstart playing with these fractions:

That looks tedious to type but there's cut and pasting involved, plusthe sensor visually shows whether your parentheses are balanced, so nogreat feat of concentration was required. And look: approaching phi(there'd need to be more of a proof).

A great story line to follow is, of course, the evolution ofcryptography. The role of computers in both decrypting and encryptingbecomes a focus, along with a bevy of little problem solvingchallenges, such as how to implement this or that algorithm.

One need not go whole hog, thinking this is a college course onnothing but crypto. Nor must the focus stay so heads down withblinders, nose in the exercises.

This ain't your grandpappy's math, wherein any history was verboten,cordoned off, side-barred. History is often front and center, such aswhen we discuss the history of tabulation (form clay tablets toHollerith machines to SQL), or the history of glyph representation(from clay tablets though unicode).